The case \(q=1\)  was intensively studied in the literature. Indeed, in [1]} Attouch, Chbani and Riahi showed that for \(q=1,\)  \(p>2,\)  \(a=1\)  and \(\alpha >3\)  the generated trajectories of (REF ) converge weakly to a minimizer of \(g\) . Further, one has \(g(x(t)-\min g=o\left(\frac{1}{t^2}\right).\)  On the other hand, if \(p<2\) , then the strong convergence result \(\liminf _{t\rightarrow +\infty }\Vert x(t)-x^*\Vert =0\)  holds, where \(x^*\)  is the element of minimum norm from \(\operatornamewithlimits{argmin}g.\)  The previous result is also true if \(p=2\)  and \(a>\frac{4}{9}\alpha (\alpha -3).\)  Similar results have been obtained in [2]} and [3]} for some second order dynamical systems with (implicit) Hessian driven damping and Tikhonov regularization term (see also [4]}, [5]}). The case \(p=2\)  seems to be critical in the sense that separates the case when one obtains fast convergence of the function values and weak convergence of the trajectories to a minimizer, and the case when the strong convergence of the trajectories to a minimizer of minimum norm is ensured. However, recently Attouch and László in [6]} succeeded to obtain both rapid convergence towards the infimal value of \(g\) , and the strong convergence of the trajectories towards the element of minimum norm of the set of minimizers of \(g.\)  More precisely, in [6]} it is shown that if \(q=1,\)  \(p=2,\)  \(a>0\)  and \(\alpha >3\)  then
\(\displaystyle g\left(x(t)\right) - \min g =O\left(\frac{1}{t^2}\right) \mbox{ as }t\rightarrow +\infty .\)  Further, the trajectory \(x\)  is bounded,
\(\displaystyle \Vert \dot{x}(t)\Vert  =O\left(\frac{1}{t}\right) \mbox{ as }t\rightarrow +\infty \) , and
there is strong convergence to the minimum norm solution \(x^*\) , i.e.
\( \liminf _{t \rightarrow \infty }{\Vert  x(t) - x^\ast \Vert } = 0.\)  A similar result has been obtained in [3]} for a second order dynamical systems with implicit Hessian driven damping.
